2y^2=6y-1

Solution for 2y^2=6y-1 equation:

Simplifying
2y2 = 6y + -1

Reorder the terms:
2y2 = -1 + 6y

Solving
2y2 = -1 + 6y

Solving for variable 'y'.

Reorder the terms:
1 + -6y + 2y2 = -1 + 6y + 1 + -6y

Reorder the terms:
1 + -6y + 2y2 = -1 + 1 + 6y + -6y

Combine like terms: -1 + 1 = 0
1 + -6y + 2y2 = 0 + 6y + -6y
1 + -6y + 2y2 = 6y + -6y

Combine like terms: 6y + -6y = 0
1 + -6y + 2y2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
0.5 + -3y + y2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -3y + -0.5 + y2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -3y + y2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -3y + y2 = 0 + -0.5
-3y + y2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-3y + y2 = -0.5

The y term is -3y.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3y + 2.25 + y2 = -0.5 + 2.25

Reorder the terms:
2.25 + -3y + y2 = -0.5 + 2.25

Combine like terms: -0.5 + 2.25 = 1.75
2.25 + -3y + y2 = 1.75

Factor a perfect square on the left side:
(y + -1.5)(y + -1.5) = 1.75

Calculate the square root of the right side: 1.322875656

Break this problem into two subproblems by setting 
(y + -1.5) equal to 1.322875656 and -1.322875656.

Subproblem 1
y + -1.5 = 1.322875656

Simplifying
y + -1.5 = 1.322875656

Reorder the terms:
-1.5 + y = 1.322875656

Solving
-1.5 + y = 1.322875656

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '1.5' to each side of the equation.
-1.5 + 1.5 + y = 1.322875656 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + y = 1.322875656 + 1.5
y = 1.322875656 + 1.5

Combine like terms: 1.322875656 + 1.5 = 2.822875656
y = 2.822875656

Simplifying
y = 2.822875656

Subproblem 2
y + -1.5 = -1.322875656

Simplifying
y + -1.5 = -1.322875656

Reorder the terms:
-1.5 + y = -1.322875656

Solving
-1.5 + y = -1.322875656

Solving for variable 'y'.

Move all terms containing y to the left, all other terms to the right.

Add '1.5' to each side of the equation.
-1.5 + 1.5 + y = -1.322875656 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + y = -1.322875656 + 1.5
y = -1.322875656 + 1.5

Combine like terms: -1.322875656 + 1.5 = 0.177124344
y = 0.177124344

Simplifying
y = 0.177124344

Solution
The solution to the problem is based on the solutions
from the subproblems.
y = {2.822875656, 0.177124344}